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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi}
%
\def\Swiech{\'Swie\kern -0.21em\lower0.79ex\hbox{$\scriptstyle\lhook$}ch}
\def\rightheadline{EJDE--1999/24\hfil Existence results for boundary problems
\hfil\folio}
\def\leftheadline{\folio\hfil M. G. Crandall, M. Kocan, P. L. Lions,
\& A. \Swiech \hfil EJDE--1999/24}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999) No.~24, pp. 1--20.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35J25, 35J60, 35J65,
35K20, 35K55, 35K60, 49L25.
\hfil\break
{\eighti Key words and phrases:} Uniformly elliptic and parabolic equations,
viscosity solutions, good solutions, exterior cone condition, barrier functions.
\hfil\break
\copyright 1999 Southwest Texas State University and
University of North Texas.\hfil\break
Submitted May 20, 1999. Published July 1, 1999.\hfil\break
} }
\bigskip\bigskip
\centerline{EXISTENCE RESULTS FOR BOUNDARY PROBLEMS FOR UNIFORMLY}
\centerline{ELLIPTIC AND PARABOLIC FULLY NONLINEAR EQUATIONS }
\medskip
\centerline{M. G. Crandall, M. Kocan, P. L. Lions, \& A. \Swiech}
\bigskip\bigskip
{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We study existence of continuous weak (viscosity) solutions of
Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly
elliptic and parabolic equations. Two types of results are obtained in
contexts where uniqueness of solutions fails or is unknown.
For equations with merely measurable coefficients we prove
solvability of the problem, while in the continuous
case we construct maximal and minimal solutions. Necessary
barriers on external cones are also constructed.
\bigskip}
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\def\text#1{\hbox{\rm\quad#1\quad}}
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\def\section#1{\equationno=0
\bigbreak\centerline{\rm \secno.\quad #1}\smallskip}
\def\eq{\global\advance\equationno by 1 \eqno(\secno.\the\equationno)}
\long\def\theorem#1#2{\removelastskip\vskip\baselineskip\noindent{\bf
Theorem #1. }{\it\ignorespaces#2} }
\long\def\proposition#1#2{\removelastskip\vskip\baselineskip\noindent{\bf
Proposition #1. }{\it\ignorespaces#2}\vskip\baselineskip}
\long\def\proof#1{\removelastskip\vskip0.5\baselineskip\noindent{\it
Proof. }\ignorespaces#1\hfill\square\vskip\baselineskip}
\long\def\proofof#1#2{\removelastskip\vskip0.5\baselineskip\noindent{\it
Proof of #1. }\ignorespaces#2\hfill\square\vskip\baselineskip}
\long\def\remark#1#2{\removelastskip\vskip\baselineskip\noindent{\bf
Remark #1. }{\ignorespaces#2}\vskip\baselineskip}
\def\secno{0}
\section{INTRODUCTION}
The main results of this note concern existence of continuous solutions of the
Dirichlet problem for fully nonlinear elliptic equations as well as parabolic
variants. To illustrate the issues, we consider the Isaacs' equation
$$
\sup_\alpha\inf_\beta\Big(-\sum_{i,j=1}^na^{\alpha,\beta}_{i,j}(x)u_{x_i,x_j}(x)
+\sum_{j=1}^n b_j^{\alpha,\beta}(x)u_{x_j}(x)
+c^{\alpha,\beta}(x)u(x)-f^{\alpha,\beta}(x)\Big)=0\eq
$$
in a bounded open domain $\Omega\subset {\Bbb R}^n$ coupled with the Dirichlet condition
$$u(x)=\psi(x)
\text{for}x\in\partial \Omega\eq$$
in two situations. In both cases the indices $\alpha, \beta$ can range over
countable sets while the symmetric
matrices $A^{\alpha,\beta}=\big(a_{i,j}^{\alpha,\beta}\big)$, the vectors
$b^{\alpha,\beta}=(b_1^{\alpha,\beta},\ldots,b_n^{\alpha,\beta})$, and the
functions $c^{\alpha,\beta}$ satisfy
$$\lambda I\le A^{\alpha,\beta}(x)\le \Lambda I\eq$$ for some positive constants
$00$. Then we can combine the proof of Proposition III.1 (2) of [15]
with the arguments from Section 5.A of [8] (which show how to relax the
standard comparison assumptions when functions are $C^{1,\alpha}$) to prove
comparison if $\omega$ in (1.4) satisfies
$\omega(r)\le Cr^\theta$ for some $\theta>(1-\bar\alpha)/(2-\bar\alpha)$. This
approach makes full use of ${\cal SC}$. If $\theta>\frac12$ then $F\in{\cal SC}$ can be
somewhat weakened, and uniqueness of solutions still holds, see [15]. For
other results in this direction see Jensen [16] and Trudinger
[31].
For the continuous coefficient linear case (0.5) existence of $W^{2,n}_{\rm loc}(\Omega)$
solutions is known. These solutions are also unique $C$-viscosity solutions -- this is
a special case of Theorem 2.10 of [4] aided by Proposition 2.9 of the
same
work. For the elliptic case, the most general results on existence of
$W^{2,n}_{\rm loc}(\Omega)$ solutions for $F\in{\cal SC}$ and convex or concave in the Hessian are
found in [28]; the parabolic analogue is in [10] (which makes use of the
results herein). These latter results rely on foundational estimates of Caffarelli
[3], Wang [32] and the contributions of Escauriaza [11].
Finally, in the good solution framework for linear equations, the results and
commentary of Safonov [25] and Cerutti, Fabes and Manselli [6]
indicate what is known on the positive side. As mentioned in the
introduction, Nadirashvili [24] shows nonuniqueness in general. See also
Safonov [27].
\def\secno{2}
\section{EXISTENCE PROOF FOR CONTINUOUS $F$}
{\it Throughout this section, the terms subsolution, supersolution
and solution mean, respectively, $C$-viscosity subsolution, $C$-viscosity supersolution
and $C$-viscosity solution (see above).}
We are ready for the proof of Theorem 1.1.
\proofof{Theorem 1.1}{The idea for the proof, once conceived,
makes the rest simple. The point is to find approximating equations
$F_\epsilon =0$
and $F^\epsilon =0$ with better dependence on $x$ and satisfying $F_\epsilon\le
F\le F^\epsilon$. Even if $F$ is linear, there is in general no linear
approximation with the properties we need. Put
$$F_\epsilon(x,r,p,X)= \min_{y\in \overline \Omega}\left(F(y,r,p,X)
+\frac1\epsilon |x-y|\right).\eq$$ Since for each fixed $y\in\overline \Omega$
$$ F(y,r,p,X) +\frac1\epsilon |x-y|$$ belongs to ${\cal SC}$ and has
the same parameters $\gamma, \lambda, \Lambda$ and
continuity in $r$ as $F$, $F_\epsilon$ shares these properties. The striking
thing is that $F_\epsilon$ is Lipschitz continuous in
$x$ with constant $1/\epsilon$ uniformly in $r,p,X$ while preserving the rest of
the structure. The operation of ``inf-convolution" used here is standard,
but this use of it is unusual.
By the
definition $F_\epsilon\le F$ (choose $y=x$ in
(2.1)). Next, $F$ is continuous on $\overline \Omega\times{\Bbb R}\times{\Bbb R}^n\times{\cal S}(n)$,
so for $R>0$ there exists $\omega_R\colon\,[0,\infty)\rightarrow[0,\infty)$ such that
$\omega_R(0+)=0$ and
$$|F(x,r,p,X)-F(y,r,p,X)|\le \omega_R(|x-y|)\text{for}|r|+|p|+\|X\|\le R\eq$$
when $x,y\in\overline \Omega$. Then for $|r|+|p|+\|X\|\le R$
$$\eqalign{F(x,r,p,X)\le& F(y,r,p,X)+\omega_R(|x-y|)\cr
&\le F(y,r,p,X)+\frac1\epsilon|x-y|+\omega_R(|x-y|)-\frac1\epsilon|x-y|,}$$
so we have
$$F_\epsilon(x,r,p,X)\le F(x,r,p,X)\le F_\epsilon(x,r,p,X)+\delta_R(\epsilon),$$
where
$$\delta_R(\epsilon)=\sup_{0< s\le {\rm diam}(\Omega)}
\left(\omega_R(s)-\frac s\epsilon\right)\rightarrow 0\text{as}\epsilon\downarrow 0.\eq$$
In particular,
$$F_\epsilon(x,r,p,X)\rightarrow F(x,r,p,X)\text{uniformly for} x\in\overline \Omega\text{and
bounded} r,p, X\eq$$
as $\epsilon\downarrow 0$. Similarly, we define
$$F^\epsilon(x,r,p,X)=\sup_{y\in\overline
\Omega}\left(F(y,r,p,X)-\frac1\epsilon|x-y|\right),\eq$$
which has all the same structure properties as $F_\epsilon$ and
satisfies
(2.4) in place of $F_\epsilon$. Moreover for $00$. The results of Miller [22], [23] provide
the existence of local barriers for the Dirichlet problem for
(3.1) under a uniform exterior cone condition. We do not know
a place in the literature where global barriers (supersolutions)
are constructed, although the case $\gamma=0$ is treated in Michael
[21] and the proof there may be modified to handle the
general case -- see Remark 3.4. We present another option, using the
flexibility of viscosity solutions.
First we recall the nature of the barriers on exterior cones for
extremal elliptic operators constructed by Miller [22],
[23]. For $n\ge2$ and $\beta\in(0,\pi)$ let
$$T_\beta=\left\{x\in{\Bbb R}^n\colon\ x_n\ge(\cos\beta)|x|\right\}$$ be
the closed circular cone of aperture $\beta$ with axis in the
direction of $-e_n$. Consider barriers of the form
$$w(x)=r^b f(\theta),\eq$$ where $r=|x|$ and
$\theta=\arccos\left(x_n/|x|\right)$. It is shown in [23], Theorem 3
and Section 7, that for every $\beta\in(0,\pi)$ there exist
$b\in(0,1)$ and $f\in C^2([0,\pi))$, depending only on
$\lambda,\Lambda,n,\beta, \gamma$, such that $f'(0)=0$ and $f>1$ on $[0,\beta]$,
so that $w$ given by (3.2) is continuous on $T_\beta$ and $C^2$ on
${\Bbb R}^n\setminus\{$closed negative $x_n$ axis$\}$ and
$$w>r^b\ \ {\rm on}\ T_\beta\setminus\{0\},\ \ w(0)=0,\eq$$ and,
crucially,
$${\cal P}^-(D^2w)-\gamma|Dw|\ge r^{b-2}\quad{\rm on}\ \ {\rm
int}(T_\beta).\eq$$
Now let $\Omega$ be a bounded domain in ${\Bbb R}^n$, $n\ge2$, satisfying a
uniform exterior cone condition. This means that there exist
$\beta\in(0,\pi)$ and
$r_0>0$ so that for every $z\in\partial\Omega$ there is a rotation
$\Theta=\Theta(z)$ such that
$$\overline\Omega\cap B_{r_0}(z)\subset z+\Theta T_\beta.\eq$$ Here,
$B_{r_0}(z)$ denotes the open ball in ${\Bbb R}^n$ of radius $r_0$ centered
at $z$. The ``local'' nature of the barriers below is due to the
possibility that $\Omega\subset z+\Theta T_\beta$ does not hold.
Setting
$w_z(x)=w(\Theta^{-1}(x-z))$, from
(3.3) we have
$$w_z(z)=0,\quad w_z(x)\ge|z-x|^b\quad{\rm on}\ \ \overline\Omega\cap
B_{r_0}(z).\eq$$ In particular, we have
$$w_z\ge (r_0)^b\quad{\rm on}\ \ \{x\in\overline\Omega
\colon\ \ |x-z|=r_0\}.\eq$$
Using (3.4) and (3.5) we arrive at
$${\cal P}^-(D^2w_z(x))-\gamma|Dw_z(x)|\ge |x-z|^{b-2}\quad{\rm for}
\ \ x\in\Omega\cap B_{r_0}(z).\eq$$
We need to extend these local barriers to global ones. Choose any
point $y\not\in\overline \Omega$ and $2r_10$
put
$$G(x)=\left(\frac1{r_1^\sigma}-\frac1{|x-y|^\sigma}\right).\eq$$ Clearly
$$G(x)\ge {2^\sigma-1\over (2r_1)^\sigma}>0\text{on}\overline \Omega\eq$$ and a
standard computation ([14]) shows that
$${\cal P}^-(D^2 G(x))-\gamma|DG(x)|\ge {\sigma\over
|x-y|^{\sigma+2}}\left((\sigma+1)\lambda-\Lambda(n-1)-\gamma|x-y|\right)>0\eq$$ on $\overline \Omega$ for
large $\sigma$ (depending only on $n,\lambda,\Lambda,\gamma$ and diam$\,(\Omega$)).
Replacing $G$ by $aG$ for a suitable $a>0$ we can achieve all of:
$$G>0\text{and} {\cal P}^-(D^2 G)-\gamma|DG|\ge\kappa \text{on}\overline \Omega\eq $$ for some
$\kappa>0$ and
$$G(x)r_0$, \cr
\min\left(G(x),w_z(x)\right) & for $x\in\overline\Omega,\ |x-z|\le r_0$\cr}\eq
$$
agrees with $w_z$ in a neighborhood of $z\in\partial \Omega$
relative to $\overline \Omega$ (in view of (3.6)
and (3.12)), agrees with $G$ on $\overline \Omega\setminus B_{r_2}(z)$ for
some $00\text{on}\overline \Omega\setminus\{z\}\eq$$ and
$${\cal P}^-(D^2W_z)-\gamma|DW_z|\ge 1\quad{\rm on}\
\ \Omega.\eq$$
The task of satisfying the boundary condition remains. Let $\psi\in
C(\partial \Omega)$ and
$$|\psi(x)-\psi(z)|\le \rho(|x-z|)\quad{\rm for\quad}x,z\in\partial \Omega,\eq$$
where $\rho(0+)=0$, so $\rho$ is a modulus of continuity for $\psi$.
In addition to ${\cal P}^-(D^2U)-\gamma|DU|\ge M$, the supersolution $U$ we
construct will satisfy
$$|U(x)-\psi(z)|\le \omega(|x-z|)\text{for}x\in\Omega,\ z\in\partial \Omega,\eq $$
where $\omega(0+)=0$; that is the boundary values are assumed
uniformly. Moreover, $\omega$ will depend only on the parameters of
the cone condition,
$\lambda,\Lambda,n,\gamma$ and the diameter of $\Omega$ (which already determine
the character of each $W_z$), and $M$ and $\rho$. For each
$\kappa>0$ and
$z\in\partial \Omega$ put
$$W_{\kappa,z}(x)=\psi(z)+\kappa +M_\kappa W_z(x),$$ where
$M_\kappa\ge M$ (guaranteeing
${\cal P}^-(D^2W_{\kappa,z})-\gamma|DW_{\kappa,z}|\ge M$ by (3.16)) is
chosen so that
$$\psi(z)+\kappa +M_\kappa W_z(x)\ge \psi(x)\text{for}x\in\partial \Omega. $$
In view of (3.17), it suffices to take
$$M_\kappa\ge \sup_{x\in\partial \Omega,x\not=z}{(\rho(|x-z|)-\kappa)^+\over
W_z(x)};$$ this may evidently be done uniformly in $z\in\partial \Omega$.
Finally we put
$$W(x)=\inf_{z\in\partial \Omega,\kappa>0}W_{\kappa,z}(x).$$ By construction $W\ge
R$. Since for all $\kappa$
$$W(x)-\psi(z)\le W_{\kappa,z}(x)-\psi(z)=\kappa +M_\kappa W_z(x) $$
and $W_z(x)$ is uniformly continuous in $x$ uniformly in $z$, we
conclude that for all $z\in\partial \Omega$ and $x\in\Omega$
$$W(x)-\psi(z)\le \omega(|z-x|)\eq$$ for some $\omega$ satisfying
$\omega(0+)=0$. We now use Remark 3.1\ -- the supersolutions of
(3.1) we have constructed imply the existence of corresponding
subsolutions of $F_\epsilon\le 0$ (or
${\cal P}^-(D^2U)-\gamma|DU|\le-M$ for an appropriate $M$) with boundary
values below $\psi$, call them $Y_{\kappa,z}$ and the supremum
$Y$. By the analogue of (3.19) for $Y$ we have
$$-\omega(|x-z|)\le Y(x)-\psi(z).\eq$$ According to [8], $U=W_*$,
the lower semicontinuous envelope of $W$, is a supersolution of
(3.1), and consequently of $F_\epsilon=0$. Similarly,
$V=Y^*$, the upper semicontinuous envelope of $Y$, is a subsolution
of
$F_\epsilon=0$. Since $V=U=\psi$ on $\partial \Omega$, comparison gives $V\le U$
and using this together with (3.19) and (3.20) yields
$$|V(x)-\psi(z)|,\ |U(x)-\psi(z)|\le \omega(|z-x|)$$ and we are done.
For convenient reference, we summarize the results of this
construction in terms of extremal equations:
\proposition{3.2}{Let $\Omega$ satisfy a uniform exterior cone
condition, $\psi\in C(\partial \Omega)$ and $M\in{\Bbb R}$. Then the problems
$${\cal P}^-(D^2u)-\gamma|Du|=M\quad{in}\ \ \Omega,\quad u=\psi\quad{on}\ \ \partial \Omega\eq$$ and
$${\cal P}^+(D^2v)+\gamma|Dv|=-M\quad{in}\ \ \Omega,\quad v=\psi\quad{on}\ \ \partial \Omega\eq$$ have
unique $C$-viscosity solutions $u,v\in C(\overline \Omega)$ satisfying $u=v=\psi$ on
$\partial \Omega$. Moreover, there is a modulus $\omega$ depending only on the
parameters of the cone condition, $\lambda,\Lambda,n,\gamma$, ${\rm
diam}\,(\Omega)$, $M$ and the modulus of continuity of $\psi$ such that
$$|u(x)-\psi(z)|,\ |v(x)-\psi(z)|\le \omega(|x-z|)\quad{for}\ \ z\in
\partial \Omega,\ x\in\Omega.$$}
Regarding the statement, recall Remark 3.1; moreover, given the
subsolutions and supersolutions exhibited above, we may assert the
existence of solutions, not only subsolutions and supersolutions.
\remark{3.3}{ Bellman equations (3.22) and (3.21) are
concave/convex in the Hessian matrix
and as such can be studied by classical methods, see Krylov
[18]. In particular, Safonov [26], Theorem 1.1, proves that under
the assumptions of Proposition 3.2, the problems (3.22) and (3.21)
have classical $C^{2,\alpha}_{_{\rm loc}}(\Omega)\cap C(\overline \Omega)$ solutions. From
uniqueness it follows that $C$-viscosity solutions $u,v$ of Proposition
3.2\ coincide with Safonov's; in particular $u,v\in C^2(\Omega)$.
However, we feel that the construction presented here is
useful. The problems (3.22) and (3.21) were solved here --
albeit in a weaker sense -- without invoking the apparatus of hard
$C^{2,\alpha}$ estimates for nonlinear equations. Our objective is to
solve equations $F_\epsilon=0$ that are not expected to have
classical solutions. $F_\epsilon=0$ is solved by Perron's method, and
for this purpose the information summarized in Proposition
3.2\ is sufficient. Corollary 3.10 of [4] extends the existence
to cover the situation when $M$ is replaced by $f\in L^p(\Omega)$ for
suitable $p$, a fact used in Section 4.
As for the parabolic result, Proposition 3.5\ below, we were
not able to locate a quotable result in literature asserting
classical solvability of (3.25) and (3.26). However, Krylov
in [18], Theorem 6.4.3, proves an analogous result in the case of
$\Omega$ satisfying an exterior sphere condition, and this generalizes to handle
the cone condition. Krylov's method consists of approximating $\Omega$ from the
inside by more regular domains; a delicate argument involving barrier
functions
and H\"older estimates is used to pass to the limit. The result of Krylov
shows that $C$-viscosity solutions of Proposition 3.5\ are $C^{2,1}$ (and more).
This follows easily by solving (CDP) classically on regular subdomains of
$\Omega\times (0,T]$ using $u$ ($v$, respectively) as the boundary data, and
invoking uniqueness.}
\remark{3.4}{ Our construction of barriers relies on the barrier
$w$ from (3.2) for a canonical unbounded cone $T_\beta$ at the origin, taken
from [22], [23]. We used uniform exterior cone
condition to obtain a local barrier $w_z$ at every $z\in\partial \Omega$;
$w_z$ is obtained by composing $w$ with an appropriate isometry
making $T_\beta$ into an exterior cone at $z$. Then $w_z$ was
extended to a global barrier $W_z$ (3.14) by means of a fixed
function $G$ in
(3.9).
Another possibility would be to first construct a barrier $W_0$ on the
exterior of a canonical bounded cone $C_{\beta,\delta}$ at the origin:
$$C_{\beta,\delta}=\left\{x\in{\Bbb R}^n\colon\ |x|\le\delta,\ x_n
\le(\cos\beta)|x|\right\}.$$ This can be accomplished by a similar
procedure as above. Namely, taking $y=-\frac\delta 2 e_n$, sufficiently
small $r_1>0$ (determined only by $\delta,\beta$) and $G$ as in
(3.9), for $\sigma$ large, depending only on $n,\lambda,\Lambda,\gamma$
and ${\rm diam}(\Omega)$ known in advance, one can guarantee
$$G>0\text{and} {\cal P}^-(D^2 G)-\gamma|DG|>0 \text{on}B_{{\rm diam}(\Omega)}(0)
\setminus C_{\beta,\delta}.$$ As in (3.14), a multiple of this $G$
combined with $w$ would produce a desired barrier $W_0$. Now one can
use uniform exterior cone condition to find for every $z\in\partial \Omega$ a
rotation $\Theta=\Theta(z)$ such that
$$\overline\Omega\cap\left( z+\Theta C_{\beta,\delta}\right)=\{z\},$$ and then
$W_z(x)=W_0\left(\Theta^{-1}(x-z)\right)$ defines a global barrier at
$z$.
Michael [21] considers barriers
$$\varphi(x)=1-e^{-Kw(x)},$$ where $w$ is Miller's barrier (3.2) and
$K>0$. [21] gives explicit recipes for $K,b$ and $f$ so that
$\varphi$ becomes a barrier for ${\cal P}^-(D^2\varphi)\ge r^{b-2}$ on the
exterior of $C_{\beta,\delta}$. This construction can be easily modified
to handle first order terms to obtain ${\cal P}^-(D^2\varphi)-\gamma|D\varphi|\ge1$
on $B_{{\rm diam}(\Omega)}(0)\setminus C_{\beta,\delta}$. This canonical
barrier $\varphi$ can be used instead of $W_0$ constructed above; note
that $\varphi$ is $C^2$ unlike $W_0$.
Finally, if $n=2$ the exterior cone condition can be replaced
by a weaker condition, see [22], [23] and Section 2.8
of [14].
}
\noindent{\bf Parabolic Case.}
The work done above renders the parabolic case simple. To construct
supersolutions, we reduce as before to the problem
$$U_t+{\cal P}^-(D^2U)-\gamma|DU|\ge M.$$ Reviewing the preceding
construction, we see that all we need will follow if we produce a
function
$W_{z,\tau}$ for each point $(z,\tau)$ of the (parabolic) boundary
$\partial \Omega\times (0,T]\cup
\overline \Omega\times\{0\}$ of
$Q$ satisfying the analogues of (3.15) and (3.16):
$$W_{z,\tau}\in C(\overline Q),\quad W_{z,\tau}(z,\tau)=0, \quad
W_{z,\tau}>0\text{on}\overline Q\setminus\{(z,\tau)\}\eq$$ and
$$(W_{z,\tau})_t+{\cal P}^-(D^2W_{z,\tau})-\gamma|DW_{z,\tau}|\ge 1\text{on} Q.\eq $$
For $\tau>0$ and $z\in\partial \Omega$ we set
$$W_{z,\tau}(x,t)=\frac1{2T}(t-\tau)^2+2W_z(x),$$ where $W_z$ was
constructed above and satisfies (3.15) and (3.16). It is clear
that (3.23) holds. Moreover, by (3.16),
$$\eqalign{(W_{z,\tau})_t+{\cal P}^-(D^2W_{z,\tau})&-\gamma|DW_{z,\tau}|\cr
=\frac1T(t-\tau)+&2\left({\cal P}^-(D^2W_z)-\gamma|DW_z|\right)\ge-1+2=1} $$ and we have
(3.24).
For $\tau=0$ and $z\in\overline \Omega$ we set
$$W_{z,0}(x,t)=At+\frac12|x-z|^2;$$ again if $A>0$ we clearly have
(3.23). Finally,
$$\eqalign{(W_{z,0})_t&+{\cal P}^-(D^2W_{z,0})-\gamma|DW_{z,0}|=\cr
&A+{\cal P}^-(I)-\gamma|x-z|=A-n\Lambda-\gamma|x-z|\ge A-n\Lambda-\gamma{\rm diam\,}(\Omega).
} $$ Thus we have (3.24) if $A=n\Lambda+\gamma{\rm diam\,}(\Omega)+1$. The
rest of the analysis follows that of the elliptic case step by step.
Here is the parabolic version of Proposition 3.2. We use the
notation introduced in Section 2.
\proposition{3.5}{Let $\Omega$ satisfy a uniform exterior cone
condition, $\psi\in C(\partial_pQ)$ and $M\in{\Bbb R}$. Then the problems
$$u_t+{\cal P}^-(D^2u)-\gamma|Du|=M\quad{in}\ \ Q,\quad u=\psi\quad{on}\ \ \partial_pQ\eq$$ and
$$v_t+{\cal P}^+(D^2v)+\gamma|Dv|=-M\quad{in}\ \ Q,\quad v=\psi\quad{on}\ \ \partial_pQ\eq$$
have unique $C$-viscosity solutions $u,v\in C(\overline{Q})$ satisfying
$u=v=\psi$ on
$\partial_pQ$. Moreover, there is a modulus $\omega$ depending only on the
parameters of the cone condition, $\lambda,\Lambda,n,\gamma,T$, ${\rm
diam}\,(\Omega)$, $M$ and the modulus of continuity of $\psi$ such that
$$|u(x,t)-\psi(z,\tau)|,\ |v(x,t)-\psi(z,\tau)|\le
\omega(|x-z|+|t-\tau|)\quad{for}\ \ (z,\tau)\in\partial_pQ,\ (x,t)\in Q.$$}
\def\secno{4}
\section{$L^p$ THEORY: GENERAL EXISTENCE}
The requirements (1.2) and (1.3)
constituting the basic structure conditions require no continuity of
$F(x, r,p,X)$ in $x$ and very little in $r$. In this section, we assume
that $F$ is merely measurable in $x$ (or $(x,t)$ in the parabolic case) while
satisfying the structure conditions for almost every $x$ (or $(x,t)$). Due to
this generality we have to impose a requirement on the $r$ dependence: for
$R>0$ there exists $\omega_R\colon\,[0,\infty)\rightarrow [0,\infty)$ such that
$\omega_R(0+)=0$ and
$$|F(x,r,p,X)-F(x,s,p,X)|\le \omega_R(|r-s|)\eq$$
for almost all $x\in\Omega$ and $|r|+|s|+|p|+\|X\|\le R$. (Obviously, if $F$ is
continuous then (4.1) automatically holds.) In the parabolic case,
$x$ is replaced by $(x,t)\in Q$.
Of course, the notions of $C$-viscosity subsolutions, etc., are no longer appropriate in
this measurable situation, and there is now a well-developed theory using
corresponding ``$L^p$-viscosity'' notions ([4], [9],
[28], [7], [10]) which is built up from the fundamental
regularity results of Caffarelli [2] (see [3]) as further developed
by Escuariaza [11] and Wang [32]. In the linear case other notions of weak
solutions were proposed, see Cerrutti, Escauriaza and Fabes [5] and Jensen
[17]; relationships between various notions of solutions are studied in
[17] and [9].
\noindent{\bf $\bf L^p$-Viscosity Notions}
In contrast to the $C$-viscosity notions recalled at the end of
Section 1, $L^p$-viscosity notions use ``test functions''
$\varphi\in W^{2,p}_{\rm loc}(\Omega)$
(functions whose distributional second derivatives are in
$L^p_{\rm loc}(\Omega)$) in the elliptic case and $\varphi\in W^{2,1,p}_{\rm loc}(Q)$ (functions
whose distributional first derivatives and second order spatial
derivatives are in $L^p_{\rm loc}(Q)$) in the parabolic case. In
addition, all subsolutions, etc., are required to be continuous.
For example, a continuous function $u$ on $\Omega$ is an $L^p$-viscosity
subsolution of
$F(x,u,Du,D^2u)=0$ if for every $\varphi\in W^{2,p}_{\rm loc}(\Omega)$ and local maximum
$\hat x$ of
$u-\varphi$ one has
$${\rm ess}\liminf_{x\rightarrow\hat x}F(x,u(x),D\varphi(x),D^2\varphi(x))\le
0;$$ equivalently, if for some $\epsilon>0$
$$F(x,u(x),D\varphi(x),D^2\varphi(x))\ge \epsilon\text{a.e.}$$ in some
neighborhood of $\hat x$, then $\hat x$ is not a local maximum of
$u-\varphi$. The corresponding notions of $L^p$-viscosity
supersolutions, $L^p$-viscosity solutions, and the parabolic versions are what
then one expects.
Consulting the literature mentioned above, one finds that there is an
equation dependent appropriate range of $p$ determined by the
parameters $\lambda, \Lambda,n,\gamma{\rm diam}(\Omega)$ (or $\gamma{\rm diam}(Q)$ for
parabolic equations). One always has $n/2

0$ and
$B_R=\{f\in L^n(\Omega)\colon\ \|f\|_{L^n(\Omega)}\le R\}$.
Then the set of all functions
$u\in C(\overline \Omega)$ such that there exists
$\psi\in{\cal C}$ and $f\in B_R$ for which $u$ is an $L^n$-viscosity solution of both
$${\cal P}^-(D^2 u)-\gamma|Du|\le f\text{\it and} -f\le
{\cal P}^+(D^2 u)+\gamma|Du| \eq$$
in $\Omega$ and $u=\psi$ on $\partial \Omega$ is
precompact in $C(\overline \Omega)$. }
\proof{According to [4] Corollary 3.10, if $(\varphi,g)\in
C(\partial \Omega)\times L^n(\Omega)$, there exist a unique $U=U(\varphi,g)
\in C(\overline \Omega)\cap W^{2,n}_{\rm loc}(\Omega)$ such that
$$ {\cal P}^-(D^2 U)-\gamma|DU|=g\text{a.e. in}\Omega\text{and} u
=\varphi\text{on}\partial \Omega. $$
We require several facts. First, there exist $p\in(n/2,n)$
depending on $\lambda, \Lambda, n, \gamma\,{\rm
diam}\,(\Omega)$
(see [13], [11], [4], [28], [1]) and $C$ such that
$$U(\varphi,g)\le \sup_{\partial \Omega}\varphi+ C\left(\int_{\Omega}(g^+)^p\right)^{1\over
p}\text{for}(\varphi,g)\in C(\partial \Omega)\times L^n(\Omega).\eq$$
Next the mapping
$$ \ C(\partial \Omega)\times L^n(\Omega)\ni(\varphi,g)\rightarrow U(\varphi,g)\text{is sublinear and
order preserving.}\eq
$$
Finally, if $u\in C(\overline \Omega)$ is an $L^n$-viscosity solution of
$$ {\cal P}^-(D^2 u)-\gamma|Du|\le f\text{in}\Omega\text{and}u=\psi\text{on}\partial \Omega, \eq$$
where $f\in L^n(\Omega)$, then
$$ u\le U(\psi,f).\eq$$
We review the genesis of these results in Remark 4.4\ below.
The second inequality of (4.6) may be restated as $w=-u$ is an
$L^n$-viscosity solution of ${\cal P}^-(D^2 w)-\gamma|Dw|\le f$, so if (4.6) holds (4.10) implies $-u\le
U(-\psi,f)$ or
$-U(-\psi,f)\le u$. All told, (4.6) and $u=\psi$ on $\partial \Omega$ yield
$$-U(-\psi,f)\le u\le U(\psi,f).\eq$$
From (4.11) and (4.7) it follows that $u$ remains bounded in $C(\overline \Omega)$
if $(\psi,f)$ remains bounded in
$C(\partial \Omega)\times L^n(\Omega)$ (or even in $C(\partial \Omega)\times L^p(\Omega)$). We now use
(4.11)
to show that $u$ assumes the boundary values $\psi$ in an equicontinuous
manner.
In this regard, let
$f_M=\max(\min(f,M),-M)$ be the standard truncation of
$f$ for
$M>0$. We note that for $f\in B_R$
$$\|f-f_M\|_{L^p(\Omega)}\le \left({\rm measure}(\{|f|>M\})\right)^{{n-p\over
np}}R\le R\left({R\over M}\right)^{{n-p\over p }},$$
which tends to 0 as $M\rightarrow \infty$ uniformly in $f\in B_R$.
Using the properties (4.8), (4.7) of $U$ we thus have
$$\eqalign{U(\psi,f)\le U(\psi,f_M)+U(0,f-f_M)&\le U(\psi,M)+C\|f-
f_M\|_{L^p(\Omega)}\cr
&\qquad\le U(\psi,M)+CR\left({R\over M}\right)^{{n-p\over p }}.}$$
According to Proposition 3.2, $U(\psi, M)$ assumes the boundary values $\psi$
in a manner controlled by the modulus of continuity of $\psi$ for fixed $M$. The
``error term" on the right above can be made as small as desired by choosing
$M$ sufficiently large, and $u\le U(\psi,f)$ thus guarantees an estimate
$u(x)-\psi(y)\le \rho(|x-y|)$ for $x\in\overline \Omega, y\in \partial \Omega$, where $\rho(0+)=0$.
Similarly,
$-U(-\psi,f)\le u$ provides control of $u-\psi$ at the boundary from below.
Finally, once
$u$ is bounded, (4.6) guarantees equi-H\"older continuity of $u$ on
compact subsets of $\Omega$ so long as $f$ remains bounded in $L^n(\Omega)$
(see, for example, [13] for a sufficiently general statement and Remark
4.7 below). The result follows.}
\remark{4.3}{ Proposition 4.2\ can be reformulated by saying that
if $u$ satisfies (4.6) and $u=\psi$ on $\partial \Omega$ then $u$ has a modulus
of continuity on $\overline\Omega$ that only depends on the parameters
of the cone
condition, $\lambda,\Lambda,n,\gamma$, ${\rm
diam}\,(\Omega)$, $R$ and the modulus of continuity of $\psi$.}
\remark{4.4}{The inequality (4.7) generalizes
the original work of Fabes and Stroock
[12] and is proved in [13] in the spirit of this
work, but
it could also be deduced from Cabr\'e [1];
its relevance in this arena was first shown by Escauriaza [11]. In fact,
the
existence of $U(\varphi,g)$ for $\gamma=0$ was proved in [11] relying on (4.7)
with
$\gamma=0$. The properties
(4.8) are a consequence of the positive homogeneity and superlinearity of
$(p,X)\rightarrow {\cal P}^-(X)-\gamma|p|$ and (4.7) ($p=n$ suffices). For example, the
superadditivity implies that $W=U(\varphi,g)-U(\hat\varphi,\hat g)$ solves
$${\cal P}^-(D^2 W )-\gamma|DW|\le g-\hat g\le 0$$ if $g\le\hat g$ and an application of (4.7)
($p=n$ suffices) then proves the order preserving property. The relation
(4.10)
given (4.9) follows upon observing that $v=u-U(\psi,f)$ is an $L^n$-viscosity solution of
${\cal P}^-(D^2 v)-\gamma|Dv|\le 0$ and the
Alexandrov-Bakelman-Pucci maximum principle for viscosity
solutions
proved in [2] ($\gamma=0$), [30], [4].
Finally, Proposition 4.2\ itself appears in
[3],
Theorem 4.14, in the situation where $\gamma=0$, $\Omega$ is a ball, and all functions $f$
appearing in
(4.6) are continuous. This proof could be adapted, with effort, to the
current
case. The current proof uses the work already done in Section 3.}
\proofof{Theorem 4.1}{First we assume that $F(x,r,p,X)$ is defined for
all
$(r,p,X)$ for all $x\in{\Bbb R}^n$ and satisfies the structure conditions
(1.2), (1.3) and (4.1) for all $x\in{\Bbb R}^n$. To
achieve this, if necessary extend
$F(x,r,p,X)$ to be ${\cal P}^-(X)-\gamma|p|$ (or ${\cal P}^+(X)+\gamma|p|$) for those
$x$'s where it was not originally defined. Now mollify $F$ in $x$:
$$F_\epsilon(x,r,p,X)={1\over\epsilon^n}
\int_{{\Bbb R}^n}\eta\left({x-y\over\epsilon}\right)F(y,r,p,X)\,dy, $$ where $\eta\in
C_0^\infty({\Bbb R}^n)$ satisfies $\eta\ge 0$ and
$\int_{{\Bbb R}^n}\eta(x)\,dx=1$. The structure conditions are preserved
under this sort of averaging, so $F_\epsilon\in{\cal SC}$.
Clearly $F_\epsilon$ satisfies (4.1), (4.4) and (4.5) as well as $F$.
Moreover, the bound (4.5) on $|F|$ gives us
$$|F_\epsilon(x,r,p,X) -F_\epsilon(y,r,p,X)|
\le \frac C\epsilon|x-y|\left(\Lambda\|X\|+\gamma|p|+\beta(r)\right)
$$
for some $C$.
Fix $f\in L^n(\Omega)$ and let $f_j\in C(\overline \Omega)$ satisfy
$$\|f_j-f\|_{L^n(\Omega)}\rightarrow 0\text{as}j\rightarrow \infty.$$
Since $F_\epsilon\in{\cal SC}$ is continuous, according to Theorem 1.1\ the problem
$$F_\epsilon(x,u,Du,D^2u)=f_j\text{in}\Omega\text{and} u=\psi\text{on}\partial \Omega\eq$$
has a $C$-viscosity solution (and hence $L^n$-viscosity solution) $u=u_{\epsilon,j}$. Clearly
$u=u_{\epsilon,j}$ also solves
$${\cal P}^-(D^2u)-\gamma|Du|+F_\epsilon(x,u,0,0)\le f_j\text{and}f_j\le
{\cal P}^+(D^2u)+\gamma|Du|+F_\epsilon(x,u,0,0).
$$ Since $F_\epsilon(x,u,0,0)\ge F_\epsilon(x,0,0,0)=0$ if $u\ge 0$, the first
relation
above and the maximum principle for
viscosity solutions implies
$$u\le
\sup_{\partial \Omega}u^++C\|f_j\|_{L^n(\Omega)}\le\sup_{\partial \Omega}\psi^+
+C\sup_j\|f_j\|_{L^n(\Omega )}
$$ and we conclude that the $u_{\epsilon,j}$ are bounded above independently
of $\epsilon, j$. Likewise, the $u_{\epsilon,j}$ are bounded below independently
of $\epsilon, j$, and hence the family is uniformly bounded. Using this
information and (4.5) for $F_\epsilon$, there exists a constant $K$ such that
$|F_\epsilon(x,u_{\epsilon,j},0,0)|\le K$ and the
$u_{\epsilon,j}$ satisfy
$${\cal P}^-(D^2u_{\epsilon,j})-\gamma|Du_{\epsilon,j}|\le g_j\text{and}-g_j\le
{\cal P}^+(D^2u_{\epsilon,j})+\gamma|Du_{\epsilon,j}|,$$ where $g_j=|f_j|+K$.
Therefore, using Proposition 4.2, there exists $\epsilon_m\downarrow 0$,
$j_m\rightarrow\infty$ such that $u_m=u_{\epsilon_m,j_m}$ converges
uniformly on $\overline \Omega$ to a limit $u$. By Theorem 3.8 of [4] this
$u$ is an $L^n$-viscosity solution of (4.3); indeed, what we need to check to
use this result is only that for $\varphi\in W^{2,n}_{\rm loc}(\Omega)$ we have
$$F_{\epsilon_m}(x,u_m(x),D\varphi(x),D^2\varphi(x)) \rightarrow
F(x,u(x),D\varphi(x),D^2\varphi(x))\eq$$ in $L^n_{\rm loc}(\Omega)$. However,
$F_\epsilon(x,r,p,X)\rightarrow F(x,r,p,X)$ whenever $x$ is a Lebesgue point of
$F(\cdot,r,p,X)$, and almost every $x$ has this property for all
$r,p,X$ by
$F\in{\cal SC}$ (see [4], page 382), which together with (4.5)
shows that (4.13) holds pointwise a.e. and (locally) dominated, hence in
$L^n_{\rm loc}(\Omega)$.}
We now turn to the parabolic analogue of Theorem 4.1. In this
case the initial boundary value problem can be rewritten as before
as
$$u_t+F(x,t,u,Du,D^2u)=f(x,t)\text{in}Q=\Omega\times (0,T],\quad
u=\psi\text{on}
\partial_pQ,\eq $$ where
$$F(x,t,0,0,0)\equiv 0.\eq$$ The proof of the
theorem below is similar to the one in the elliptic case and is
therefore omitted, save for the remarks to follow.
\theorem{4.5}{Let $F\in{\cal SC}$ satisfy (4.1) and (4.15), let $f\in
L^{n+1}(Q)$, $\psi\in C(\partial_pQ)$ and let $\Omega$ satisfy a uniform exterior cone
condition. Then (4.14) has an $L^{n+1}$-viscosity solution.}
The parabolic
version of the compactness result Proposition 4.2\ is
\proposition{4.6}{Let $\Omega$ satisfy a uniform exterior cone
condition and ${\cal C}\subset C(\partial_pQ)$ be compact, $R>0$ and
$B_R=\{f\in L^{n+1}(Q)\colon\ \|f\|_{L^{n+1}(Q)}\le R\}$. Then the set of all functions $u\in
C(\overline Q)$
such that there exists
$ \psi \in{\cal C}$ and $f\in B_R$ for which $u$ is an $L^{n+1}$-viscosity solution
of both
$$u_t+{\cal P}^-(D^2u)-\gamma|Du|\le f\text{\it and} -f\le u_t+{\cal P}^+(D^2u)+\gamma|Du| \eq$$
in $Q$ and $u=\psi$ on $\partial_pQ$ is
precompact in $C(\overline Q)$. }
A version of the maximum principle and an existence result sufficient for
the proof of this proposition is given in [7]. The interior H\"older
continuity is established in [10], Section 5. The limit theorem needed to complete
the proof of Theorem 4.5\ is proved in [10], Section 6.
\remark{4.7}{ The proofs of the existence results above do not require
full Propositions 4.2\ and 4.6\ but rather their versions
with $B_R$ replaced by $B_R\cap C(\Omega)$ (or $B_R\cap C(Q)$). In this
case, the proofs of the versions of the maximum principles and equi-H\"older
continuity results found in Caffarelli [2], Trudinger [30] (elliptic case),
and Wang [32] (parabolic case) could be used. This leaves aside (4.7), upon
which we have commented. The parabolic analogue is proved in [7]. The proofs of
the various maximum principles sketched in [7] might interest the reader in
any case.}
\remark{4.8}{ We note again, for emphasis, that Theorems 4.1\ and
4.5\ are also true if $n$ and $n+1$ are replaced by $p$ in (parameter
dependent) appropriate ranges of the form $n-\delta < p $ and $n+1-\delta